the number of maximal subgroups and probabilistic...

IntroductionLarge characteristically simple sections of a group

The upper bound

The number of maximal subgroups andprobabilistic generation of finite groups1

Ramón Esteban-Romero12

1Departament de MatemàtiquesUniversitat de València

2Institut Universitari de Matemàtica Pura i AplicadaUniversitat Politècnica de València

Ischia (NA), March, 2018 / Ischia Group Theory 2018

1Joint work with A. Ballester-Bolinches, P. Jiménez-Seral and H. MengRamón Esteban-Romero Maximal subgroups and probabilistic generation


The upper bound

Outline

1 IntroductionProbabilistic generationBounds of Jaikin-Zapirain and PyberAims of this talk

2 Large characteristically simple sections of a groupPrimitive groupsNew results

3 The upper boundOur boundsConsequencesExamples

Ramón Esteban-Romero Maximal subgroups and probabilistic generation


The upper bound

Probabilistic generationBounds of Jaikin-Zapirain and PyberAims of this talk

Outline






The upper bound


IntroductionProbabilistic generation

All groups in this talk will be finite.

Motivating question

Let G be a d-generated group. How many elements one shouldexpect to choose uniformly and randomly to generate G? ε(G)



The upper bound



Netto, 1880: The probability that a randomly chosen pairof elements of Alt(n) generates Alt(n) tends to1 as n→∞ (conjecture).The probability that a randomly chosen pair ofelements of Sym(n) generates Sym(n) tendsto 3/4 as n→∞ (conjecture).

E. Netto.The theory of substitutions and its applications to Algebra.Register Publ. Co., Inland Press, Ann Arbor, Michigan, USA, 1892.Translation from the original (1880) in German.



The upper bound



Dixon, 1969: Netto’s conjecture is true: the proportion ofgenerating pairs for Alt(n) or Sym(n) isgreater than 1− 2/(ln ln n)2 for sufficientlylarge n.He conjectures that the same happens forsimple groups.

J. D. Dixon.The probability of generating the symmetric group.Math. Z., 110(3):199–205, 1969.



The upper bound



Kantor, Lubotzky, 1990; Liebeck, Shalev, 1995: Dixon’sconjecture is valid: If G is almost simple with socleS, the probability that a pair of elements of Ggenerates a subgroup containing S tends to 1 as|G| → ∞.

W. M. Kantor and A. Lubotzky.The probability of generating a finite classical group.Geom. Dedicata, 36(1):67–87, 1990.

M. W. Liebeck and A. Shalev.The probability of generating a finite simple group.Geom. Dedicata, 56(1):103–113, 1995.



The upper bound



Pomerance, 2001: If G is abelian, then

ε(G) ≤ d(G) + σ,

where σ = 2.118456563 . . . is obtained fromRiemann zeta function (best possible for abeliangroups) and d(G) is the minimum number ofgenerators of G.

C. Pomerance.The expected number of random elements to generate a finite abeliangroup.Period. Math. Hungar., 43(1-2):191–198, 2001.



The upper bound



Definition (Pak)G group.ν(G): least positive integer k such that G is generated by krandom elements with probability at least 1/e.

Theorem (Pak)

1eε(G) ≤ ν(G) ≤ e

e− 1ε(G).

I. Pak.On probability of generating a finite group.Preprint, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.7319.


http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.7319

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.7319


The upper bound



Since

〈x1, . . . , xk 〉 6= G ⇐⇒ ∃M l G such that 〈x1, . . . , xk 〉 ≤ M,

the maximal subgroups are relevant in the scope ofprobabilistic generation. In fact,

Prob(〈x1, . . . , xk 〉 ≤ M) =k∏

i=1

Prob(xi ∈ M)

=

(|M||G|

)k

=1

|G : M|k

The number mn(G) of maximal subgroups of G of a given indexn is also relevant.



The upper bound



Theorem (Lubotzky, 2002)If G is a group with r chief factors in a given chief series, then

mn(G) ≤ r(r + nd(G))n2 ≤ r2nd(G)+2.

Furthermore,

ν(G) ≤ 1 + log log|G|log i(G)

+ max(

d(G),log log|G|log i(G)

)+ 2.02,

i(G): smallest index of a proper subgroup of Glog: logarithm to the base 2

A. Lubotzky.The expected number of random elements to generate a finite group.J. Algebra, 257:452–459, 2002.



The upper bound



Theorem (Detomi and Lucchini, 2003)There exists a constant c such that, for any group G,ν(G) ≤ bd(G) + c logλ(G)c if λ(G) > 1, otherwise,ν(G) ≤ bd(G) + cc, where λ(G) denotes the number ofnon-Frattini chief factors in a given chief series of G.

bxc: defect integer part of x .

E. Detomi and A. Lucchini.Crowns and factorization of the probabilistic zeta function ofa finite group.J. Algebra, 265(2):651–668, 2003.



The upper bound



DefinitionFor a group G, let us call

M(G) = maxn≥2

logn mn(G) = maxn≥2

log mn(G)

log n.

logn x : logarithm to the base n of x , that is,logn x = log x/ log n = ln x/ ln n.

Theorem (Lubotzky, 2002)

M(G)− 3.5 ≤ ν(G) ≤M(G) + 2.02.



The upper bound


Outline






The upper bound


IntroductionBounds of Jaikin-Zapirain and Pyber

DefinitionLet G be a group, A a characteristically simple group.

rkA(G): largest number r such that G has a normal sectionthat is the direct product of r non-Frattini chieffactors of G that are isomorphic (not necessarilyG-isomorphic) to A.

l(G): least degree of a faithful transitive permutationrepresentation of G, i.e., the smallest index of acore-free subgroup of G.

A. Jaikin-Zapirain and L. Pyber.Random generation of finite and profinite groups and groupenumeration.Ann. Math., 173:769–814, 2011.



The upper bound



Theorem (Jaikin-Zapirain, Pyber, 2011, Theorem 1)

There exist two absolute constants 0 < α < β such that forevery group G we have

α

(d(G) + max

A

log rkA(G)

log l(A)

)< ν(G)

< β d(G) + maxA

log rkA(G)

log l(A)

,

where A runs through the non-abelian chief factors of G.

The max on the RHS is 0 if G is soluble (JZ-P, privatecommunications).



The upper bound



Theorem (Jaikin-Zapirain, Pyber, 2011, Theorem 9.5)

Let G be a d-generated group. Then

max

d ,maxn≥5

log rkn(G)

c7 log n− 4≤ ν(G)

≤ cd + maxn≥5

log max1, rkn(G)log n

+ 3,

where c and c7 are two absolute constants.

rkn(G): maximum of rkA(G), where A runs over the non-abeliancharacteristically simple groups A with l(A) ≤ n.



The upper bound


Outline






The upper bound


IntroductionAims of this talk

Aims:To give an interpretation of the invariant rkA(G) for anon-abelian characteristically simple group A.To improve the upper bound for mn(G) and, hence, forν(G).To estimate the values of the constants in Theorem 9.5 ofJaikin-Zapirain and Pyber, 2011.



The upper bound

Primitive groupsNew results

Outline






The upper bound


Large characteristically simple sections of a groupPrimitive groups

DefinitionA primitive group is a group with a core-free maximal subgroup.

If M is a maximal subgroup of G, then M/MG is a core-freemaximal subgroup of G/MG and so G/MG is primitive.



The upper bound



Theorem (Baer, 1957)Let G be a primitive group and let U be a core-free maximalsubgroup of G. Exactly one of the following statements holds:

1 Soc(G) = S is a self-centralising abelian minimal normalsubgroup of G, G = US and U ∩ S = 1.

2 Soc(G) = S is a non-abelian minimal normal subgroup ofG, G = US. In this case, CG(S) = 1.

3 Soc(G) = A× B, where A and B are the two uniqueminimal normal subgroups of G, G = AU = BU andA ∩ U = B ∩ U = A ∩ B = 1. In this case, A = CG(B),B = CG(A), and A ∼= B ∼= AB ∩ U are non-abelian.



The upper bound



Theorem (Baer, 1957)Let G be a primitive group and let U be a core-free maximalsubgroup of G. Exactly one of the following statements holds:

1 Soc(G) = S is a self-centralising abelian minimal normalsubgroup of G, G = US and U ∩ S = 1 (type 1).

2 Soc(G) = S is a non-abelian minimal normal subgroup ofG, G = US. In this case, CG(S) = 1 (type 2).

3 Soc(G) = A× B, where A and B are the two uniqueminimal normal subgroups of G, G = AU = BU andA ∩ U = B ∩ U = A ∩ B = 1. In this case, A = CG(B),B = CG(A), and A ∼= B ∼= AB ∩ U are non-abelian (type 3).



The upper bound



DefinitionThe primitive group [H/K ] ∗G associated with a chief factorH/K of G is:

1 the semidirect product [H/K ](G/CG(H/K )

)if H/K is

abelian, or2 the quotient group G/CG(H/K ) if H/K is non-abelian.



The upper bound


Outline






The upper bound


Large characteristically simple sections of a groupNew results

Theorem

Let G be a monolithic primitive group in which B = Soc(G) isnon-abelian. Then G/B has no chief factors isomorphic to B.



The upper bound


Large characteristically simple sections of a groupNew results

Theorem B

Let A be a non-abelian chief factor of a group G and supposethat in a given chief series of G there are k chief factorsisomorphic to A. Then there exist two normal subgroups C andR of G such that R ≤ C and C/R is isomorphic to a directproduct of k minimal normal subgroups of G/R isomorphic to A.

This can be extended to non-Frattini abelian chief factors.In particular, rkA(G) is the number of chief factors of Gisomorphic to A in a given chief series of G.The proof depends on the precrown associated to asupplemented chief factor and a maximal subgroupsupplementing it.



The upper bound

Our boundsConsequencesExamples

Outline






The upper bound


The upper boundOur bounds

Definition

Let G be a group and let n ∈ N, n > 1. We denote by crAn (G) thenumber of crowns associated to complemented abelian chieffactors of order n of G, that is, the number of G-isomorphismclasses of complemented abelian chief factors of G.

Clearly, crAn (G) = 0 unless n is a power of a prime.This invariant concerns type 1 maximal subgroups (G/MGprimitive of type 1).



The upper bound



DefinitionLet n ∈ N. The symbol rksn(G) denotes the number ofnon-abelian chief factors A in a given chief series of G suchthat the associated primitive group [A] ∗G has a core-freemaximal subgroup of index n.

This invariant concerns type 2 maximal subgroups.



The upper bound



DefinitionLet n ∈ N. The symbol rkon(G) denotes the number ofnon-abelian chief factors A in a given chief series of G suchthat |A| = n.

DefinitionLet n ∈ N. The symbol rkomn(G) denotes the maximum of thenumbers rkA(G) for A running over the isomorphism types ofnon-abelian chief factors of G with |A| = n.

These invariants concern type 3 maximal subgroups.The non-abelian chief factors A of G of order n fall into atmost two isomorphism classes.



The upper bound



Theorem A

Let G be a d-generated non-trivial group. Then

maxd ,maxA

log rkA(G)

2 log l(A)− 2.63 ≤ ν(G) ≤ η(G),

where in the maximum on the left hand side, A runs over theisomorphism classes of non-abelian chief factors in a givenchief series of G and η(G) is a function bounded by a linearcombination of d and the maxima of logn crAn (G), logn rksn(G),logn rkon(G), and logn rkomn(G).



The upper bound



Lemma (Borovik, Pyber, Shalev, 1996)

The number g(n) of isomorphism classes of non-abelian simplesubgroups of Sym(n) for n ≥ 5 is O(n).

We precise the value O(n):

LemmaThe number g(n) of isomorphism classes of non-abelian simplegroups of Sym(n) for n ≥ 5 is at most 4.89n + 1 141.33.

A. V. Borovik, L. Pyber, and A. Shalev.Maximal subgroups in finite and profinite groups.Trans. Amer. Math. Soc., 348(9):3745–3761, 1996.



The upper bound



LemmaThe number s(n) of isomorphism classes of minimal normalsubgroups of primitive groups of type 2 with a core-freemaximal subgroup of index n satisfies the inequalitys(n) ≤ n1.266.

Note that rksn(G) ≤ s(n)rkn(G).



The upper bound



TheoremLet U be a maximal subgroup of type 1 of a d-generated groupG and let n = |G : U|. Then the number of maximal subgroupsM of G such that Soc(G/MG) is G-isomorphic to Soc(G/UG) isless than or equal to

nd − n|H1(G/C,A)|q − 1

,

where A = C/UG is the unique minimal normal subgroup ofG/UG and q = |EndG/C(A)|. In particular, this number is lessthan nd .



The upper bound



Corollary (number of type 1 maximal subgroups)

The number of type 1 maximal subgroups M of index n = pr ofa d-generated group G is less than or equal to (nd − 1)crAn (G).

We use arguments of Dalla Volta and Lucchini (1998) andresults of Gaschütz (1959).

F. Dalla Volta and A. Lucchini.Finite groups that need more generators than any properquotient.J. Austral. Math. Soc. Ser. A, 64(1):82–91, 1998.

W. Gaschütz.Die Eulersche Funktion endlicher auflösbarer Gruppen.Illinois J. Math., 3(4):469–476, 1959.



The upper bound



Theorem (number of type 2 maximal subgroups)

Let G be a group and let n ∈ N. The number of maximalsubgroups of G of type 2 and index n is bounded by rksn(G)n2.



The upper bound



Theorem (number of type 3 maximal subgroups)

Let G be a d-generated group and let n ∈ N which is a power ofthe order of a non-abelian simple group. The number ofmaximal subgroups of G of type 3 and index n is bounded by

n2 min

nd ,rkomn(G)− 1

2

rkon(G).

This result depends on the study of the crowns associated tonon-abelian chief factors, since the minimal normal subgroupsof G/MG are G-connected.



The upper bound



Theorem

1 If n ∈ T (power of a prime), then (types 1 and 2)mn(G) ≤ (nd − 1)crAn (G) + n2rksn(G)

≤ 2 maxndcrAn (G),n2rksn(G).2 If n ∈ S (power of the order of a non-abelian simple group),

then (types 2 and 3)

mn(G) ≤ n2rksn(G) + n2 min

nd ,rkom(G)− 1

2

rkon(G).

≤ 2n2 max

rksn(G),min

nd ,rkomn(G)− 1

2

rkon(G)

.

3 If n /∈ S ∪ T, then mn(G) ≤ n2rksn(G) (type 2).



The upper bound



Theorem A

Let G be a d-generated non-trivial group. Then, for

η(G) := max

d + 2.02 + maxlogn 2 + logn crAn (G),

4.02 + maxlogn 2 + logn rksn(G),4.02 + max

mind + logn 2, logn rkomn(G)

+ logn rkon(G),

we have thatν(G) ≤ η(G).



The upper bound


Outline






The upper bound


The upper boundConsequences

The following result is Corollary 7.3 of Jaikin-Zapirain andPyber, 2011, written in a stronger form.

Theorem (cf. Jaikin-Zapirain and Pyber, 2011, Corollary 7.3)Let G be a d-generated group. There exists a constant c6 suchthat the number of irreducible G-modules of size n is at most

nc6d max1, rkn(G).



The upper bound



This result can be used to obtain Theorem 9.5 ofJaikin-Zapirain and Pyber, 2011 from our results, because

logn crAn (G) ≤ c6d + logn max1, rkn(G),logn rksn(G) ≤ 1.266 + logn max1, rkn(G),logn rkon(G) ≤ log2 2 + logn max1, rkn(G).

It follows that

ν(G) ≤ (c6 + 1)d + 3.02 + max logn max1, rkn(G).



The upper bound



The bound of Jaikin-Zapirain and Pyber depends on a constantdefined as a linear combination of constants that in many casesare known to exist, but no explicit values have been given forthem. We precise the values in Theorem 9.5 of Jaikin-Zapirainand Pyber, 2011.

Theorem

Let G be a d-generated group. Then

η(G) ≤ cd + maxn≥5

log max1, rkn(G)log n

+ 3,

where c = 375.06.



The upper bound



This result depends on the previously mentioned result.

Corollary (cf. Jaikin-Zapirain, Pyber, 2011, Corollary 7.3)Let G be a d-generated group. There exists a constant C6 suchthat the number of irreducible G-modules of size n is at most

max1, rkn(G)nC6d .

This constant can be taken to be 374.06.



The upper bound



A slightly different approach shows:

Theorem

The number of non-equivalent irreducible G-modules ofsize n = pr , where p is a prime and r ∈ N, is at most

nminc6d+k6+logn max1,rkn(G),dr,

where c6 = 183.034 and k6 = 74.



The upper bound


Outline






The upper bound


The upper boundExamples

Corollary

Let G be a d-generated group with no abelian chief factors.Then ν(G) ≤ 4.289 + d + maxn≥5 logn max1, rkn(G).



The upper bound



ConstructionG d-generated primitive group of type 1.Ω = ordered generating d-tuples of G = Ω1 ∪ · · · ∪ Ωr ,with Ωi orbits of the action of Aut(G) on Ω,(gi1, . . . ,gid ) ∈ Ωi , 1 ≤ i ≤ r ,gj =

∏ri=1 gij ∈ Gr , 1 ≤ j ≤ d .

G = 〈g1, . . . ,gd〉 is a subdirect product of Gr and G has assocle a direct product of all faithful and irreducible modulesfor G whose primitive group is isomorphic to G.This construction can be extended to many d-generatedprimitive groups with isomorphic socles.



The upper bound



There are 3 isomorphism classes of 2-generated primitivegroups of type 1 with socle of order 8: G1 = [C3

2 ]C7,G2 = [C3

2 ][C7]C3, G3 = [C32 ]GL3(2).

We can construct a 2-generated group S with all possiblecrowns of abelian chief factors of order 8.

crA3 (S) = 1, crA7 (S) = 9, crA8 (S) = 146, rkGL3(2)(S) = 57.

Bound of Jaikin-Zapirain and Pyber:ν(S) ≤ 3 + 2c + log7 57 with3 + 2c + log7 57 ≥ 5.07 + 2c ≈ 755.198.Bound of Theorem A: ν(S) ≤ 6.75.


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